Superconducting spin qubits, realized as Yu-Shiba-Rusinov spin-doublet states in quantum-dot-superconductor systems, represent a cornerstone of current research in quantum technologies. We analyze these ground states of quantum impurities in superconducting pseudogap systems, namely systems with a pseudogap tunneling density of states $\rho (ϵ)\sim |ϵ{|}^r$ for energies $|ϵ|\gg \Delta$ ($\Delta$ being a $s$-wave pairing potential). For $r=1$, these hosts are realized as Dirac materials (graphene or 3D topological insulator surfaces) in proximity to conventional superconductors, or as $d+is$ superconductors. Using effective field theory and numerical renormalization group, we map the phase diagram against the pseudogap exponent $r>0$ and particle-hole symmetry-breaking perturbations. At particle-hole symmetry, increasing $r$ also increases the critical value, $J_c$, of the Kondo coupling that triggers the transition from spin doublet to singlet. Unlike the gapless pseudogap Kondo systems, numerical and analytical evidence suggest that Andreev reflection stabilizes a singlet ground state at $J$ for all $r>0$. Breaking particle-hole symmetry —by potential scattering or chemical potential— eventually restores the transition at lower $J_c$. Our results indicate that coupling to superconducting hosts with large pseudogap exponents enhances the stability of spin qubits at large Kondo coupling.